Abstract In micromechanics, the stress–force–fabric (SFF) relationship is referred to as an analytical expres-sion linking the stress state of a granular material with microparameters on contact forces and material fabric.This paper employs the SFF relationship and discrete element modelling to investigate the micromechanicsof fabric, force and strength anisotropies in two-dimensional granular materials. The development of the SFFrelationship is briefly summarized while more attention is placed on the strength anisotropy and deformationnon-coaxiality. Due to the presence of initial anisotropy, a granular material demonstrates a different behaviourwhen the loading direction relative to the direction of the material fabric varies. Specimens may go throughvarious paths to reach the same critical state at which the fabric and force anisotropies are coaxial with theloading direction. The critical state of anisotropic granular material has been found to be independent of theinitial fabric. The fabric anisotropy and the force anisotropy approach their critical magnitudes at the criticalstate. The particle-scale data obtained from discrete element simulations of anisotropic materials show that inmonotonic loading, the principal force direction quickly becomes coaxial with the loading direction (i.e. thestrain increment direction as applied). However, material fabric directions differ from the loading direction andthey only tend to be coaxial at a very large shear strain. The degree of force anisotropy is in general larger thanthat of fabric anisotropy. In comparison with the limited variation in the degree of force anisotropy with varyingloading directions, the fabric anisotropy adapts in a much slower pace and demonstrates wider disparity in theevolution in the magnitude of fabric anisotropy. The difference in the fabric anisotropy evolution has a moresignificant contribution to strength anisotropy than that of force anisotropy. There are two key parameters thatcontrol the degree of deformation non-coaxiality in granular materials subjected to monotonic shearing: theratio between the degrees of fabric anisotropy and that of force anisotropy and the angle between the principalfabric direction and the applied loading direction.
Presented at the 8th European Solid Mechanics Conference in the Graz University of Technology, Austria, 9–13 July 2012.
X. Li (B)Fluids and Particle Processes Group, Manufacturing and Process Technologies Research Division,Faculty of Engineering, The University of Nottingham, University Park, Nottingham, NG7 2RD, UKE-mail: [email protected].: +44-115-9514167Fax: +44-115-9513898URL: www.nottingham.ac.uk/∼evzxl
H.-S. YuNottingham Centre for Geomechanics, Materials, Mechanics and Structures Research Division,Faculty of Engineering, The University of Nottingham, University Park, Nottingham, NG7 2RD, UKE-mail: [email protected].: +44-115-8466884Fax: +44-115-9513898URL: http://www.nottingham.ac.uk/engineering/departments/civeng/people/hai-sui.yu
2346 X. Li, H.-S. Yu
1 Introduction
Granular materials have anisotropic structures either formed during natural geological processes or resultingfrom various in situ loading conditions. Extensive experimental and in situ data have shown the significant effectof material anisotropy on the observed stress–strain responses. For example, a cross-anisotropic sand sample,which can sustain a high shear stress when loaded vertically, may yield and even collapse when the sameloading is applied horizontally [21,46]. Strength anisotropy is of key engineering importance in estimating thestability of infrastructures and has therefore attracted much research interest. Extensive efforts of experimentaltesting and constitutive modelling have been made to better estimate the strength and deformation of anisotropicgranular soils [2,3,5,6,9,19,22,27,29,32,34,42]. Experiments have been carried out by preparing and testingspecimens of different tilting angles or loading a soil specimen along various directions [20,29,30,45]. Inaddition, numerical simulations using discrete element methods [8] are reported and found in qualitativeagreement with laboratory observations [22,31,41].
Another interesting phenomenon associated with anisotropic soil behaviour is deformation non-coaxiality,which is defined as the non-coincidence between the principal stress directions and the principal strain incre-ment directions [1,11,36,44,47]. It was firstly observed in Roscoe et al. [36] and later widely reported forsoil tests involving rotation of stress directions [13], for example, using a hollow cylindrical apparatus. Theangle between the principal stress direction and the principal direction of plastic strain rate is referred to as thedegree of non-coaxiality.
Material anisotropy is believed to be the main reason causing strength anisotropy and deformation non-coaxiality. However, this issue may not be fully addressed in continuum mechanics by only considering agranulate soil as an equivalent continuum. Multi-scale approaches treating a granular material as an assemblyof discrete particles have been developed over the past few decades. It has become increasingly popular asparticle-scale information has been made more accessible nowadays [10,14,18]. The fabric tensor has beenproposed to characterize material structure and incorporated in constitutive model development to better capturethe material behaviour [27,33,40,42]. Challenges remain on establishing the correlation between the proposedfabric tensors and the key characteristics of material constitutive behaviours.
Rothenburg and Selvadurai [39] were among the first to introduce Fourier series in the description of thedirectional variation of contact normal density. They further derived the stress–force–fabric (SFF) relationshipfor two-dimensional assemblies consisting of discs [37], and later extended the expression to two-dimensionalelliptical-shaped particles [38] and three-dimensional ellipsoidal particles with anisotropy tensors [35]. TheSFF relationship proposed by Rothenberg and his co-workers expressed the macroscopic stress tensor withan explicit statistical description in terms of microscopic parameters. It is an analytical relationship providingmicromechanical insight into the stress state of granular materials.
Recently, Li and Yu [24] employed the theory of directional statistics to study the statistics of particle-scaleinformation. They revisited the assumptions made in Rothenburg and his co-workers’ derivation and proposeda more general form of the SFF relationship using tensor multiplication. The form of polynomial expansions indirection n proposed in Kanatani [16,17] was followed to approximate the directional distributions of contactnormal density, mean contact vector and mean contact force. The least square error criterion was employed todetermine the tensorial coefficients, i.e. the direction tensors. These direction tensors are macroscopic variablesrepresenting the particle-scale statistics. The SFF relationship and the theory of directional statistics are usefultools in studying the micromechanics and exploring the micro–macro relationships of granular materials.Following a brief summary of this new development, this paper applies them to study the micromechanics ofanisotropic granular materials.
In a study of the dependence of granular material behaviour on initial fabric and loading directions, Liand Yu [22] prepared and tested two anisotropic specimens consisting of non-spherical particles using a two-dimensional commercial discrete element package, Particle Flow Code in Two Dimensions (PFC2D) [15].Observations were made on strength anisotropy and deformation non-coaxiality for both specimens. Multi-scale data obtained from Li and Yu [22] will be used here, where appropriate, to facilitate our discussion inthis paper.
2 A summary of the SFF relationship
In this paper, an Einstein summation convention is adopted for repeated subscripts unless indicated otherwise.
Fabric, force and strength anisotropies in granular materials: a micromechanical insight 2347
2.1 The derivation of the SFF relationship
Treating a granular material as an assembly of granular particles with only point contact interactions andvolume forces, the macro-stress tensor can be evaluated from the tensor product of contact forces f c
i andcontact vectors vc
i as follows:
σi j = 1
V
∑
c∈V
vci f c
j , (1)
in which σi j stands for the average stress over volume V [4,7,12,26,28,39,43]. To be consistent with the signconvention in soil mechanics, a contact vector is defined here as the vector pointing from the contact point tothe particle centre. Equation (1) was established based on Newton’s second law of motion and is subject to noconstraints on particle or sample geometries. For each internal contact point between the particles P and Q,there is always a pair of action and reaction forces in association with two contact vectors pointing from thecontact point to each of the particle centres. They are accounted as two contacts and contribute to Eq. (1) astwo separate terms.
In granular mechanics, contact direction is important. Denoting the contact normal direction as n, a unitdirection vector normal to the particle surface at the contact point, the terms on the right-hand side of Eq. (1)can be grouped according to their contact normal directions, leading to
σi j = 1
V
∑
�
⟨vi f j
⟩ |n�M(n) = M
V
∑
�
ec(n)⟨vi f j
⟩ |n��, (2)
where� represents the unit circle in two-dimensional spaces (D = 2) and the unit sphere in three-dimensionalspaces (D = 3). ∗|n denotes the value of variable * in direction n, and 〈∗〉 |n denotes the average value ofall terms of ∗ sharing the same contact normal direction n. The total number of contacts is denoted as M ,and �M(n) represents the number of contacts whose normal directions fall into the stereo-angle element�� centred at direction n. ec(n) = �M(n)/�� is the probability density function of contact normals. Theaverage number of contacts per particle is ω = M/N , where N is the total number of particles. In the case ofthermodynamic limit, ω approaches a limit, i.e. lim
N→∞ M/N = ω. It is referred to as the coordination number,
an index characterizing the packing density.When �� → 0, the transition leads to an expression of the stress tensor in terms of integration over all
stereo-angles as follows:
σi j = ωN
V
∮
�
ec(n)⟨vi f j
⟩ |nd�, (3)
where d� is an elementary solid angle. Equation (3) is a directional integration over the product of the contactnormal probability density ec(n) and the joint product
⟨vi f j
⟩ |n.Equation (3) involves the joint product
⟨vi f j
⟩ |n within the integration. In general,⟨vi f j
⟩ |n = 〈vi 〉 |n⟨f j⟩ |n,
where 〈vi 〉 |n and⟨f j⟩ |n denote the mean contact vector and the mean contact force along direction n, respec-
tively. The statistical dependence between them can be investigated by comparing the directional distribution of⟨vi f j
⟩ |n, 〈vi 〉 |n⟨f j⟩ |n [24]. The observation supports the assumption that the statistical dependence between
contact vectors and contact forces can be considered isotropic. This is the first simplification made in thederivation of the SFF relationship.
Simplification 1 The effect of the statistical dependence between contact vectors and contact forces could betaken into account by approximating:
⟨vi f j
⟩ |n = ς 〈vi 〉 |n⟨f j⟩ |n (4)
with ς a direction independent scalar.
Applying the directional statistic theory [16,23], directional distributions, such as ec(n), 〈vi 〉 |n and 〈 fi 〉 |n,can be approximated in terms of polynomials of unit directional vector n. Normally, only a limited numberof terms is necessary for approximation. Statistical analyses were carried out to process the microscale dataobtained in 2D DEM simulations reported in [22]. The results suggested that it is sufficient to approximate thedirectional distributions of contact normal density, mean contact forces and mean contact vectors with up to 2nd,3rd and 1st ranks of power terms of direction vector n [24]. This supports the following three simplifications.
2348 X. Li, H.-S. Yu
Simplification 2 The contact normal density ec(n) can be sufficiently approximated with up to 2nd ranks ofpower terms of the direction vector n as follows:
Ec(n) = 1
E0
(D0 + Dc
i1i2ni1ni2
), (5)
where Dci1i2
is the 2nd rank deviatoric direction tensor for contact normal density. It is a deviatoric andsymmetric tensor.
Simplification 3 The mean contact force 〈f〉 |n can be sufficiently approximated with up to 3rd ranks of powerterms of the direction vector n as:
Fj (n) = f0
(n j + G f
ji1ni1 + G f
ji1i2i3ni1ni2 ni3
), (6)
where G fji1
and G fji1i2i3
are the deviatoric direction tensors for the 1st rank and 3rd rank polynomial terms
in the approximation. They are deviatoric tensors, and G fji1i2i3
is symmetric with respect to the subscriptsi1, i2, i3.
Simplification 4 The mean contact vector 〈v〉 |n can be sufficiently approximated with up to 1st ranks ofpower terms of the direction vector n as:
Vj (n) = v0
(n j + Gv
j i1ni1
), (7)
where Gvj i1
is the deviatoric direction tensors for the first rank polynomial term in the approximation. It isagain a deviatoric tensor.
With the above simplifications, the directional integration in Eq. (3) can be achieved in terms of tensormultiplication, leading to:
σi j = ωN
Vςv0 f0
⎡
⎢⎢⎢⎢⎣
α2
(δi j + G f
ji + Gvi j + G f
jl1Gv
il1
)
+ 23α4
(Dc
i j + Dcim1
G fjm1
+ Dcim1
Gvjm1
+ Dcl1m1
Gvil1
G fjm1
)
+ 25α6
(Dc
k1k2G f
jik1k2+ Dc
k1k2Gv
il1G f
jl1k1k2
)
⎤
⎥⎥⎥⎥⎦, (8)
where α2n =⎧⎨
⎩
2nCn22n , D = 2
12n+1 , D = 3
and nCk stands for the number of k-combinations of an n-element set [24].
Equation (8) is valid for both two-dimensional spaces and three-dimensional spaces for granular materials ofvarious particle shapes, as long as the above four simplifications are considered reasonable.
2.2 Two-dimensional SFF relationship in terms of Fourier Expansions
2.2.1 Contact normal density ec(n)
In two-dimensional conditions, the unit direction vector n can be represented as n = (cos θ, sin θ) in terms ofthe angle θ in the given coordinate system. For contact normal density, the deviatoric direction tensor of thesecond-order power term can be represented as
Dci1i2
= dc2
(cosφc
2 sin φc2
sin φc2 − cosφc
2
), (9)
where dc2 denotes the magnitude of directional variation, and φc
2/2 indicates the preferred principal direction,as exemplified in Fig. 1. Simplification 2 implies that the directional distributions of contact normal densitycould be approximated as
Ec(n) = 1
2π
[1 + dc
2 cos(2θ − φc
2
)]. (10)
Fabric, force and strength anisotropies in granular materials: a micromechanical insight 2349
Isotropic term
0.05
0.10
0.15
0.20
0.25
0.00
0.05
0.10
0.15
0.20
0.25
1 2 π
Anisotropic term
0.02
0.04
0.06
0.08
0.00
0.02
0.04
0.06
0.08
-
-
+
+
2 2cφ
2 2cd π
=+0.05
0.10
0.15
0.20
0.25
0
30
6090
120
150
180
210
240270
300
330
0.00
0.05
0.10
0.15
0.20
0.25
2
2
0.36
130.95
c
c
d
φ=
=
DEM data Eq. (10)
Co
nta
ct N
orm
al D
ensi
ty
Fig. 1 Directional distribution of contact normal density ec(n)
It is the summation of an isotropic component and an anisotropic component as shown in Fig. 1.Now we will make use of the particle-scale information obtained from Li and Yu [22] for illustration.
The approximation using Eq. (10) is plotted in Fig. 1 together with the data directly obtained from DEMsimulation. When the initially anisotropically specimen was sheared at α = 45◦ up to 2 % of deviatoric strain,it is calculated that dc
2 = 0.36, φc2 = 131◦, as indicated in the figure.
2.2.2 Mean contact force 〈f〉 |nFor mean contact forces, Li and Yu [24] showed that Eq. (6) is sufficient and the two-dimensional deviatoricdirection tensors for the 1st rank and 3rd rank terms in the approximation take the form
G fji1
= B f1
(cosβ f
1 sin β f1
sin β f1 − cosβ f
1
), G f
ji111 = A f3
(cosα f
3 sin α f3
− sin α f3 cosα f
3
), (11)
where B f1 and A f
3 denote the magnitudes of directional variation, β f1 and α f
3 give information on the preferabledirections.
Simplification 3 suggests that the mean contact force could be approximated as
〈f〉 |n = f0
⎡
⎣(
cos θ
sin θ
)+ B f
1
⎛
⎝cos(θ − β
f1
)
− sin(θ − β
f1
)
⎞
⎠+ A f3
⎛
⎝cos(
3θ − αf
3
)
sin(
3θ − αf
3
)
⎞
⎠
⎤
⎦ . (12)
Decomposing the mean contact force into a normal component and a tangential component, we have⟨f n ⟩ |θ = f0
[1 + B f
1 cos(
2θ − βf
1
)+ A f
3 cos(
2θ − αf
3
)]= f0
[1 + C f
n cos(
2θ − φf
n
)], (13)
⟨f t ⟩ |θ = f0
[−B f
1 sin(
2θ − βf
1
)+ A f
3 sin(
2θ − αf
3
)]= − f0C f
t sin(
2θ − φf
t
), (14)
where C fn =
√B f 2
1 + A f 23 + 2B f
1 A f3 cos
(β
f1 − α
f3
), tan φ f
n =(
B f1 sin β f
1 + A f3 sin α f
3
)/(B f
1 cosβ f1
+ A f3 cosα f
3
), C f
t =√
B f 21 + A f 2
3 − 2B f1 A f
3 cos(β
f1 − α
f3
)and tan φ f
t =(
B f1 sin β f
1 − A f3 sin α f
3
)/
(B f
1 cosβ f1 − A f
3 cosα f3
). The mean normal contact force and the mean tangential contact force are sinusoidal
functions with period π , while the magnitudes and phase angles for normal and tangential forces may notnecessarily be the same. The parameters in Eqs. (13) and (14) are indicated in Fig. 2 to exemplify directionaldistributions of normal and tangential contact forces.
When the initially anisotropic specimen is sheared up to 2 % of deviatoric strain, the deviatoric directiontensor for contact forces is calculated with f0 = 0.078N , B f
1 = 0.35, β f1 = 98◦, A f
3 = 0.09, α f3 = 106◦
and c fn = 0.44, φ
fn = 100◦, c f
t = 0.26, φ ft = 96◦. Substituting these parameters into Eqs. (13) and (14)
gives the approximation of mean contact forces. They are plotted in Fig. 2 together with actual DEM data forcomparison.
2350 X. Li, H.-S. Yu
Anisotropic term: 1st rank
0.01
0.02
0.03
0.00
0.01
0.02
0.03
1 2fβ
0 1ff B
-
-
+
+
Isotropic term
0.020.040.060.080.10
0.000.020.040.060.080.10
0f
0 3ff A
Anisotropic term: 3rd rank
0.003
0.006
0.009
0.012
0.000
0.003
0.006
0.009
0.012
3 2fα
-
-
+
+
Anisotropic term: 1st rank
0.01
0.02
0.03
0.00
0.01
0.02
0.03
1 2fβ
0 1ff B
-
-
+ +
Anisotropic term: 3rd rank
0.003
0.006
0.009
0.012
0.000
0.003
0.006
0.009
0.012
0 1ff B
--
+
+
3 2fα
(a)
(b)
++
=
=+
0306090120150
180210240270300
330360
0.05
0.10
0.15
0.00
0.05
0.10
0.15
0 0.078N
0.44
99.99
fn
fn
f
c
φ
=
=
=
No
rmal
Co
nta
ct F
orc
e (N
)
DEM data Eq.(13)
0f
0f
nf C
2fnφ
0f
tf c
0306090120150
180210240270300330
360
0.005
0.010
0.015
0.020
0.025
0.000
0.005
0.010
0.015
0.020
0.025
0 0.078N
0.26
95.53
ft
fn
f
c
φ
=
=
=
Tan
gen
tial
Co
nta
ct F
orc
e (N
) DEM data Eq.(14)
-
+
-
+
2ftφ
Fig. 2 Approximation of mean contact force 〈f〉 |n with Eqs. (13) and (14). a Mean normal contact force. b Mean tangentialcontact force
2.2.3 Mean contact vector 〈v〉 |nThe two-dimensional deviatoric direction tensors for the 1st rank term in the approximation of mean contactvectors can be expressed as
Gvj i1
= Bv1
(cosβv1 sin βv1sin βv1 − cosβv1
), (15)
where Bv1 denotes the magnitude of directional variation, βv1 indicates the preferable direction. The magnitudesand the phase angles for the normal and tangential components are equal because only the 1st rank term isused here for approximation.
Simplification 4 suggests that the directional distributions of mean contact vectors can be sufficientlyapproximated as
〈v〉 |θ = v0
[(cos θ
sin θ
)+ Bv1
(cos(θ − βv1
)
− sin(θ − βv1
))]
. (16)
Fabric, force and strength anisotropies in granular materials: a micromechanical insight 2351
(a)
(b)
=+
Isotropic term
0.1
0.2
0.3
0.0
0.1
0.2
0.3
0v
Anisotropic term: 1st rank
0.005
0.010
0.015
0.000
0.005
0.010
0.015
1 2vβ
0 1vv B-
-
+ +
Anisotropic term: 1st rank
0.005
0.010
0.015
0.000
0.005
0.010
0.015
+
0 1vv B
-
+1 2vβ-
=0.005
0.010
0.015
0.000
0.005
0.010
0.015
0 1vv B
0
1
1
0.193mm
0.053
6.71
v
v
v
B
β
=
=
=
DEM data Eq.(18)
Tan
gen
tial
Co
nta
ct V
ecto
r (m
m)
-+
+-
1 2vβ
0.1
0.2
0.3
0.0
0.1
0.2
0.3
0
1
1
0.193mm
0.053
6.71
v
v
v
B
β
=
=
=
DEM data Eq.(17)
No
rmal
Co
nta
ct V
ecto
r (m
m)
0 1vv B
1 2vβ
Fig. 3 Approximation of mean contact vector 〈v〉 |n with Eqs. (17) and (18). a Mean normal contact vector. b Mean tangentialcontact vector
Decomposing the mean contact vector 〈v〉 |n into a normal component and a tangential component leads to:
⟨vn ⟩ |θ = v0
[1 + Bv1 cos
(2θ − βv1
)], (17)
⟨vt ⟩ |θ = v0
[−Bv1 sin(2θ − βv1
)]. (18)
When the initially anisotropic specimen is sheared up to 2 % of deviatoric strain, it has been calculated thatv0 = 0.193 mm, Bv1 = 0.053, β f
1 = 6.71◦. The approximations are plotted in Fig. 3 together with theinformation collected from DEM simulation for comparison. The disparity in the mean tangential contactvector is negligible due to the fact that the anisotropic magnitude is extremely low.
2.2.4 Simplified SFF in two dimensions
The stress tensor expression can be further simplified by invoking the symmetry in the Cauchy stress tensor,i.e. σ12 = σ21. Note that the contact normal density, mean normal contact force and mean contact vector in alldirections are nonnegative, and the magnitudes of the direction tensors are of limited range (between 0 and 1).Under most conditions, the triple products of the high rank terms of the three direction tensor are insignificantand can be ignored. Noticing that Dc
i1i2, Gv
j i1and G f
ji1are symmetric and deviatoric tensors, Li and Yu [24]
proposed a simplified form of the SFF relationship in two-dimensional spaces:
σi j = ωN
2Vςv0 f0
[(1 + h) δi j + G f
ji + Gvi j + 1
2Dc
i j
], (19)
where h = 12 G f
il1Gv
il1+ 1
4
(Dc
ik1G f
ik1+ Dc
ik1Gv
ik1
)+ 1
8 Dck1k2
G fiik1k2
is a scalar accounting for the contribution
from the joint products. It is interesting to note that G fji1i2i3
does not appear directly in Eq. (19), but contributes
only to the coefficient h through the joint product Dck1k2
G fiik1k2
.
2352 X. Li, H.-S. Yu
With the expressions of deviatoric direction tensors given in Eqs. (9), (11) and (15), the SFF relationshipin two dimensions can be expressed, in a component form, as follows:
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
σ11 = ωN2V ςv0 f0
[(1 + h)+
(B f
1 cosβ f1 + Bv1 cosβv1 + 1
2 dc2 cosφc
2
)],
σ12 = σ21 = ωN2V ςv0 f0
[B f
1 sin β f1 + Bv1 sin βv1 + 1
2 dc2 sin φc
2
],
σ22 = ωN2V ςv0 f0
[(1 + h)−
(B f
1 cosβ f1 + Bv1 cosβv1 + 1
2 dc2 cosφc
2
)].
(20)
Hence, we have the expression of the mean normal stress as:
p = ωN
2Vς (1 + h) v0 f0 (21)
and the normalized deviatoric stress tensor as:
ηi j = σi j
p− δi j = 1
1 + h
(G f
ji + Gvi j + 1
2Dc
i j
). (22)
It is symmetric and deviatoric and can be expressed in the form
ηi j = η
2
(cos θσ sin θσ
sin θσ − cos θσ
), (23)
where η is the material stress ratio and θσ /2 denotes the principal stress direction. The stress ratio η = q/p isdefined as the ratio of deviatoric stress q to mean normal stress p. In 2D spaces, the mean normal stress andthe deviatoric stress are defined as p = (σ1 + σ2)/2 and q = σ1 − σ2, respectively, where σ1 and σ2 are themajor and minor principal stresses.
3 Dependence of material stress on fabric anisotropy
3.1 A combined fabric tensor definition
The normalized deviatoric stress tensor, represented in Eq. (23) in terms of the stress ratio η and the principalstress direction θσ , can be determined by anisotropy magnitudes of material fabric and contact forces (particleinteractions), and their principal directions as seen in Eq. (22). In this relationship, Dc
i j and Gvi j are the
two deviatoric direction tensors describing particle-scale geometries (i.e. the internal structure of granularmaterials). Dc
i j characterizes the anisotropy in the contact normal density and is loading sensitive. Gvi j defines
the anisotropy in the mean contact vector and has a close correlation with the particle orientation anisotropy. Wecan conveniently define a single fabric anisotropy tensor Ci j to reflect the combined influence of anisotropiesof both contact normal density and mean contact vector as follows:
Ci j = Gvi j + 1
2Dc
i j . (24)
Since Gvi j and Dc
i j are deviatoric and symmetric tensors, the combined fabric anisotropy tensor Ci j is thereforealso deviatoric and symmetric, expressed as
Ci j = Gvi j + 1
2Dc
i j = �
(cosψ sinψ
sinψ − cosψ
), (25)
where � =√(
Bv1)2 + (dc
2
)2/4 + Bv1 dc2 cos
(βv1 − φc
2
)gives information on the magnitude of the combined
fabric anisotropy and ψ/2 indicates the principal fabric direction with tanψ = [Bv1 sin βv1 +(dc2 sin φc
2
)/2] /
[Bv1 cosβv1 +(dc
2 cosφc2
)/2].
Fabric, force and strength anisotropies in granular materials: a micromechanical insight 2353
With the single fabric anisotropy tensor defined by Eq. (24), the normalized deviatoric stress tensor becomes
ηi j = 1
1 + h
(G f
ji + Cvi j
). (26)
For an isotropic specimen, the fabric anisotropy tensor Cvi j = 0. The principal stress direction is always
coaxial with the force anisotropy direction, and the stress ratio is the same irrespective of the loading direction.However, when the material fabric is anisotropic, the non-coincidence between the principal fabric directionand force direction leads to a variation in material stress ratio and a possible deviation between the principalstress direction and the force anisotropy direction.
3.2 Fabric anisotropy and stress ratio
With Eqs. (11), (24) and (26), the stress ratio can be found as:
η = 2
1 + h
(�+ B f
1
)⎡
⎢⎣1 − 2�/B f
1(�/B f
1 + 1)2
(1 − cos
(ψ − β
f1
))⎤
⎥⎦
1/2
. (27)
This equation shows that stress ratio mobilized in a granular materials under shearing is determined by themagnitudes of fabric and force anisotropies,� and B f
1 . It is also affected by the non-coincidence between the
principal directions of force and fabric anisotropies(ψ − β
f1
)/2. When fabric anisotropy and force anisotropy
are coaxial, i.e.(ψ − β
f1
)/2 = 0◦, the stress ratio is maximal and is equal to η0 = 2
(�+ B f
1
)/(1 + h).
When there is a non-coincidence between the two principal directions(ψ − β
f1
)/2, the stress ratio becomes
η = κη0, where
κ =⎡
⎢⎣1 − 2�/B f
1(�/B f
1 + 1)2
(1 − cos
(ψ − β
f1
))⎤
⎥⎦
1/2
. (28)
The value of κ is calculated from the ratio �/B f1 and the deviation in the two principal directions(
ψ − βf
1
)/2. In Fig. 4, κ is plotted against
(ψ − β
f1
)/2 at different values of �/B f
1 . It is shown that κ
decreases with increasing(ψ − β
f1
)/2, indicating a smaller stress ratio when the fabric direction rotates away
from the force direction towards being normal to it. The decrease in stress ratio becomes more significant whenthe ratio �/B f
1 increases, suggesting that a larger fabric anisotropy causes a higher strength anisotropy. The
stress ratio becomes zero, when �/B f1 = 1 and
(ψ − β
f1
)/2 = 90◦.
3.3 Fabric anisotropy and principal stress direction
Using Eqs. (11), (24) and (26), we can also determine the principal stress direction θσ /2. Denote θ/2 =(θσ − β
f1
)/2 as the angle between the principal stress direction and the principal direction of force anisotropy.
We have θσ = θ + βf
1 , where the angle θ can be determined from
tan θ =(�/B f
1
)sin(ψ − β
f1
)
(�/B f
1
)cos(ψ − β
f1
)+ 1
(29)
2354 X. Li, H.-S. Yu
0 10 20 30 40 50 60 70 80 900
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
Deviation of phase angles ( ψ-βf1)/2 (o)
κ
Δ/Bf1=0.0
Δ/Bf1=0.2
Δ/Bf1=0.5
Δ/Bf1=0.8
Δ/Bf1=1.0
Δ/Bf1=1.25
Fig. 4 The normalized stress ratio
0 10 20 30 40 50 60 70 80 900
10
20
30
40
50
60
Deviation of phase angles (ψ-βf1)/2 (o)
Ang
le θ
/2 (
o )
Δ/Bf1=0.0
Δ/Bf1=0.2
Δ/Bf1=0.5
Δ/Bf1=0.8
Δ/Bf1=1.0
Δ/Bf1=1.25
Fig. 5 The principal stress direction (after Li and Yu [25])
Figure 5 plots the principal stress direction in terms of θ/2 at different values of�/B f1 and
(ψ − β
f1
)/2.
When the fabric anisotropy is much smaller than the force anisotropy, i.e. the ratio �/B f1 is small, the
angle between the principal stress direction and the principal force direction is small with the maximum
occurring around(ψ − β
f1
)/2 = 45◦. When the fabric anisotropy is larger and becomes comparable with
force anisotropy, the angle between the principal stress direction and the principal force direction increases and
the maximal value skews towards higher(ψ − β
f1
)/2 values. With �/B f
1 = 1, the principal stress is always
in the middle between the principal force direction and the principal fabric direction, i.e. θ =(ψ + β
f1
)/2.
When the ratio increases further so that�/B f1 > 1, the principal stress direction becomes closer to the fabric
anisotropy direction than the force anisotropy direction.
4 Numerical simulation results
In a study on material anisotropy, Li and Yu [22] prepared and tested two anisotropic specimens using discreteelement modelling. Particles used in the simulations were formed by clumping two equal-sized discs together.The distance between the centres of the two discs was equal to 1.5 times the disc radius, r . The particle sizewas uniformly distributed within (0.2, 0.6) mm in terms of equivalent diameter, and the disc thickness wast = 0.2 mm. The contact law included two constant stiffness models (normal and tangential) and a slip model.In the simulations, the elastic models were linear, and the particle stiffness was set to be kn = ks = 105 N/m.The coefficient of friction was 0.5.
Fabric, force and strength anisotropies in granular materials: a micromechanical insight 2355
Each specimen was sheared in varying loading direction from vertical to horizontal with 15◦ intervals.One was the initially anisotropic sample prepared using the deposition method. The void ratio of the preparedsample was 0.204 at pc = 1,000 kPa. The other was the preloaded sample, prepared by shearing the initiallyanisotropic sample vertically up to 25 % deviatoric strain and then unloading to isotropic stress state. The voidratio of the preloaded sample was 0.222 with the mean normal stress pc = 1,000 kPa. The loading directionwas denoted by the angleα, the deviation to the horizontal direction, i.e., x1 axis direction. Loading is applied ina strain-controlled mode with the principal strain direction fixed. Shearing was carried out at constant confiningstresses.
Further details of the numerical simulation are available in the paper [22]. Statistical analyses were carriedout on the particle and contact information. Together with the insights provided by the theoretical SFF rela-tionship, some of these numerical results are used here to facilitate our discussions on strength anisotropy anddeformation non-coaxiality.
4.1 Fabric and force anisotropy
In Eq. (19), the contributions from the joint products of direction tensors have been taken into account in termsof the coefficient h, which affects the mean normal stress defined in Eq. (21) as well as the stress ratio definedin Eq. (27). For the two series of numerical simulations reported in [22], the coefficient h was found to increaseas a result of the developments in fabric and force anisotropies and when their principal directions becomemore coaxial as shear continues. However throughout the whole shearing process, its magnitude remains smalland its influence is considered secondary. Hence, in the following, we will mainly focus our discussion onthe fabric anisotropy tensor Ci j and the force anisotropy tensor G f
ji and consider the micromechanics ofanisotropic granular materials in terms of their evolutions.
The evolution of fabric anisotropy and contact force anisotropy for the two series of tests are plottedin Figs. 6 and 7 in terms of the anisotropy magnitudes, � and B f
1 , and the phase angles, ψ/2 and β f1 /2,
respectively.The two specimens have initially anisotropic structures, evidenced by the nonzero values of fabric
anisotropy magnitudes. The principal fabric directions at the initial state were both vertical with ψ/2 = 90◦.Before shearing, the stress state is isotropy with η = 0. At this state, G f
ji = −Cvi j from Eq. (26). That is to
say, the force anisotropy and the fabric anisotropy were of equal magnitudes, while their principal directionswere normal to each other, as supported by the data points in the figures.
The fabric anisotropy for the preloaded specimen was larger than that of the initially anisotropic specimen atthe initial state due to preloading. However in comparison with the preloaded specimen, the initially anisotropicspecimen quickly developed higher fabric anisotropy in the loading direction. As for contact force anisotropy,the initially anisotropic specimen developed a high anisotropy magnitude at very low strain level, while theincrease in the preload specimen was less rapid. This is possibly due to the difference in their void ratios.
Once shearing started, the principal direction of force anisotropy adjusted almost instantaneously to theimposed loading direction, while the principal direction of fabric anisotropy adjusted much slower. For all thesimulations, the principal directions of force anisotropy were observed to be very close to the loading direction.The magnitudes of force anisotropy differed slightly when the loading direction changed, as seen in Fig. 7.As for fabric anisotropy, when its principal direction was non-coaxial with the loading direction, the principalfabric direction rotated and gradually approached the loading direction. During this process, the evolutionsof the magnitudes of fabric anisotropy were observed to be different. When the loading was coaxial with thefabric anisotropy, the fabric anisotropy kept increasing upon shearing. The magnitudes of fabric anisotropybecame smaller when the loading direction deviated further away from the principal fabric direction, and evenexperienced a temporary decrease in the case of α = 0◦, as seen in Fig. 6.
At large strain levels, the specimens approached critical states. Due to the differences in the relative directionbetween initial fabric and loading, material fabric goes through various paths to reach the ultimate state at whichboth the fabric anisotropy and the force anisotropy approached their respective critical magnitudes. Despite thedifferences in the loading directions and hence the early evolutions, the two anisotropic specimens approachedthe same critical state characterized by macroscopically the critical stress ratio ηc = 0.85, and microscopically,
the critical material fabric anisotropy�c = 0.18 and the critical force anisotropy(
B f1
)
c= 0.27. At the critical
state, the directions of both fabric anisotropy and force anisotropy were coaxial with the loading direction.
Fig. 6 Fabric anisotropy. a Initially anisotropic specimen. b Preloaded specimen
Material behaviour became coaxial, and the stress ratios for different loading directions became equal at thecritical state.
4.2 Strength anisotropy
It should be noted that the stress ratio η = q/p, when reaching a peak value during shearing, can be convertedinto a peak angle of internal friction and is generally used to define shear strength of a granular materials[47]. Equation (27) provides an analytical expression of the mobilized stress ratio and can be used to study themicromechanics of material shear strength. For completeness, the evolutions of the mobilized stress ratio forthe two series of simulations are shown in Fig. 8. The solid symbols are the stress measured on the specimenboundaries, while the hollow symbols are the predictions from Eq. (27). It is evident that the two sets of dataare in good agreement and hence validate the use of Eq. (27). The specimens developed a larger stress ratiowhen the loading direction was closer to the principal fabric direction.
The stress ratio is given in terms of microparameters as defined in Eq. (27). Apart from the secondary
parameter h, the stress ratio is mainly a function of the sum of the two magnitudes(�+ B f
1
), the ratio of
the two magnitudes �/B f1 and the angle between the principal directions
(ψ − β
f1
)/2. Figure 9 plots the
evolutions of(�+ B f
1
)for the two series of tests. Since the value of h is small, the plot of
(�+ B f
1
)is
very informative in showing the variation of the maximal stress ratio as if the two anisotropies are coaxial.(�+ B f
1
)seems to be the dominant influential parameter governing the stress ratio, supported by the close
similarity between Figs. 8 and 9.
Fabric, force and strength anisotropies in granular materials: a micromechanical insight 2357
Fig. 8 Observations on strength anisotropy (after Li and Yu [25]) a Initially anisotropic specimen, b Preloaded specimen
The stress ratio is affected by the ratio of fabric anisotropy to force anisotropy �/B f1 as a result of the
non-coincidence between their principal directions. The non-coincidence between fabric anisotropy and forceanisotropy can potentially further reduce material stress ratio by a factor of κ as evidenced in Eq. (29). Figure 4
plots κ in terms of the anisotropy ratio�/B f1 and the angle
(ψ − β
f1
)/2. In general, the magnitude of fabric
anisotropy is smaller than that of force anisotropy (�/B f1 < 1). In this case, the larger the angle
(ψ − β
f1
)/2,
the larger the ratio �/B f1 , the smaller the mobilized stress ratio becomes.
)a Initially anisotropic specimen, b Preloaded specimen
0.0
0.2
0.4
0.6
0.8
1.0
1.2
η 2(Δ+Bf
1) 2κ(Δ+Bf
1)
Loading Direction (o)
Str
ess
Rat
io
0 15 30 45 60 75 90 0 15 30 45 60 75 900.0
0.2
0.4
0.6
0.8
1.0
1.2
η 2(Δ+Bf
1) 2κ(Δ+Bf
1)
Loading Direction (o)
Str
ess
Rat
io
(a) (b)
Fig. 10 Stress ratios at 2 % deviatoric strain level. a Initially anisotropic specimen, b Preloaded specimen
Figure 10 plots information of the stress ratio η together with 2(�+ B f
1
)and 2κ
(�+ B f
1
)at 2 % of
deviatoric strains, a strain level close to the occurrence of the peak stress ratio (i.e. material strength) in the densesamples. For comparison, information of both the initially anisotropy specimen and the preloaded specimenis presented. The figure gives clear evidence of strength characteristics of anisotropic granular materials. Theobservation was shown to be in qualitative agreement with the strength anisotropy of Portway sand [5] using
a hollow cylinder apparatus. Slight differences between data points of 2κ(�+ B f
1
)and η were observed as
a result of neglecting the joint product term h.
2(�+ B f
1
)and 2κ
(�+ B f
1
)are shown almost coincident with each other, suggesting that κ is close
to 1 at 2 % of strain level. This can be explained based on the evolution of the ratio �/B f1 and the angle(
ψ − βf
1
)/2. When the loading direction is further away from the initial principal fabric direction, the angle
between the two anisotropies(ψ − β
f1
)/2 is larger, while the ratio of�/B f
1 gets smaller. Even though force
anisotropy and fabric anisotropy are still non-coaxial, the angle(ψ − β
f1
)/2 has reduced to less than 20◦ at
2 % of strain level. And the influence of non-coincidence between force and fabric is not notable.Peak stress ratio develops at a strain level around 2 % or even larger, when the angle
(ψ − β
f1
)/2 gets even
smaller. Hence, at the strain level of material peak strength, κ is very close to 1. This suggests that materialstrength anisotropy is mainly consequential to the differences in the sum of the magnitudes of fabric anisotropyand force anisotropy
(�+ B f
1
).
Information of the fabric anisotropy degree � and the force anisotropy degree B f1 is presented in Figs. 6
and 7. When the specimen was loaded in different directions, the degrees of force anisotropy differed butin a limited range, as seen in Fig. 7, while for fabric anisotropy, the evolutions of the magnitudes of fabric
Fabric, force and strength anisotropies in granular materials: a micromechanical insight 2359
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Fabric anisotropy Δ Force anisotropy Bf
1
η/2
Loading Direction (o)
An
iso
tro
py
Mag
nit
ud
es
0 15 30 45 60 75 90 0 15 30 45 60 75 900.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8 Fabric anisotropy Δ Force anisotropy Bf
1
η/2
Loading Direction (o)
An
iso
tro
py
Mag
nit
ud
es
(a) (b)
Fig. 11 Anisotropic magnitudes at 2 % of deviatoric strain level. a Initially anisotropic specimen, b Preloaded specimen
anisotropy were observed to be very sensitive to the loading direction. A large and monotonic increase wasobserved for the case α = 90◦ when the loading was coaxial with the fabric anisotropy. The magnitudes offabric anisotropy became smaller, when the loading direction deviated further away from the principal fabricdirection, and even experienced a temporary decrease in the case α = 0◦, as seen in Fig. 6. In a word, thevariation of fabric anisotropy due to varying loading direction is more significant than that of force anisotropy.
Figure 11 compares the magnitudes of fabric anisotropy � and force anisotropy B f1 at 2 % of deviatoric
strain, for both the initially anisotropy specimen and the preloaded specimen. η/2 is also plotted in the same
figure, bearing in mind that(�+ B f
1
)is the governing parameter on η/2. It is observed that the magnitude
of force anisotropy is in general larger than that of fabric anisotropy. This explains why the peak stress ratio isobserved at around the same level of peak force anisotropy. Combining information in Figs. 6 and 7, we cansee that after the peak, the continuous increase in the fabric anisotropy compensates partially the sharp dropin force anisotropy. The stress ratio hence softens gradually towards the critical stress ratio.
4.3 Deformation non-coaxiality
Supported by the observations in Fig. 7, we can assume that in monotonic shearing, the direction of forceanisotropy is coaxial with the strain increment (i.e. loading direction). As a result, the degree of non-coaxialityis equal to the deviation of the principal stress direction from that of force anisotropy, i.e. θ/2. As can be seenfrom Eq. (29), the degree of non-coaxiality is determined by the ratio�/B f
1 and the angle between contact force
and material fabric(ψ − β
f1
)/2. The dependence of the degree of non-coaxiality on�/B f
1 and(ψ − β
f1
)/2
is shown in Fig. 5. The evolution of�/B f1 and
(ψ − β
f1
)/2 can be inferred by comparing the fabric and force
anisotropies presented in Figs. 6 and 7. When the loading direction varies from horizontal (α = 0◦) to vertical
(α = 90◦), the ratio �/B f1 becomes larger, while the deviation in phase angle
(ψ − β
f1
)/2 gets smaller.
For the initially anisotropic specimen, the contact force anisotropy increases rapidly upon shearing, andtherefore, the ratio �/B f
1 is even smaller in comparison with that of the preloaded specimen. Referring to
Fig. 5, when the ratio �/B f1 is small, the degree of non-coaxiality is expected to small and the maximal
degree of non-coaxiality appears at around α = 45◦. This explains our observations on the deformation non-coaxiality of the initially anisotropic specimen, as reproduced in Fig. 12a. In Fig. 12, the solid symbols are thedata measured on the specimen boundaries, while the hollow symbols are the predictions from Eq. (29).
For the preloaded sample, the anisotropy in contact force increases less rapidly and the ratio�/B f1 is higher
than that of the initially anisotropic specimen, in particular in the small strain level. When the loading directionvaries from horizontal (α = 0◦) to vertical (α = 90◦), the ratio�/B f
1 is observed to increase even further. At
2% of shear strain, the values of �/B f1 in general lie within the range of (0.4, 0.6) except when α = 0◦ and
α = 90◦. As can be seen from Fig. 5, we now expect a much more remarkable degree of non-coaxiality (up to20◦). Larger degrees of non-coaxiality appear when the loading direction is closer to the horizontal direction.
2360 X. Li, H.-S. Yu
α=90o α=75o α=60o α=45o
α=30o α=15o α=0o
Principal Direction (o)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Str
ess
Rat
io, η
(a)
α=90o α=75o α=60o α=45o
α=30o α=15o α=0o
Principal Direction (o)
-10 0 10 20 30 40 50 60 70 80 90 100
-10 0 10 20 30 40 50 60 70 80 90 1000.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Str
ess
Rat
io, η
(b)
Fig. 12 Observations on deformation non-coaxiality (after Li and Yu [25]). a Initially anisotropic specimen. b Preloaded specimen
This explains the observation on the deformation non-coaxiality for the preloaded specimen, as reproduced inFig. 12b.
In a summary, despite that the initially anisotropic sample and the preloaded sample are of similaranisotropic degrees before shearing, the ratio �/B f
1 in the preloaded specimen is much higher than thatin the initially anisotropic specimen. As a result, it was observed for the preloaded sample that non-coaxialitybetween the principal directions of stress and strain increment was significant, but for the initially anisotropicsample, it was negligible.
5 Conclusions
The SFF relationship has been established using the directional statistical theory. It gives an analytical expres-sion of the material stress tensor in terms of fabric and force direction tensors. In two dimensions, the SFF canbe written in a very concise form, as given in Eq. (19). Based on the newly derived SFF and the DEM simu-lation results obtained from [22], we have investigated the micromechanics of anisotropic granular materialsby focusing on fabric, force and strength anisotropies and their evolutions during shearing. The key findingsof this investigation are given below:
• A combined, single, fabric anisotropy tensor Ci j can be used to take into account the effect of both contactnormal density anisotropy and contact vector anisotropy.
• For the materials used in the DEM simulations, the magnitude of force anisotropy is generally larger thanthat of fabric anisotropy. In monotonic loading, once shearing started, the principal direction of forceanisotropy adjusts almost instantaneously to the imposed loading direction, while the fabric anisotropyonly gradually approaches the loading direction.
• At large strain levels, the specimens approach critical states characterized macroscopically by the criticalstress ratio ηc = 0.85, and microscopically by the critical degree of fabric anisotropy �c = 0.18 and
the critical degree of force anisotropy(
B f1
)
c= 0.27. At the critical states, the directions of both fabric
anisotropy and force anisotropy are coaxial with the loading direction.
Fabric, force and strength anisotropies in granular materials: a micromechanical insight 2361
• Material strength anisotropy has been shown to be mainly due to the differences in the sum of the degrees
of fabric anisotropy and force anisotropy(�+ B f
1
). Between these two, the variation of fabric anisotropy
due to varying loading direction contributes more to strength anisotropy than that of force anisotropy. Itis also potentially affected by the non-coincidence between the principal directions of force and fabric
anisotropies(ψ − β
f1
)/2 depending on the ratio of these two magnitudes�/B f
1 as demonstrated in Fig. 4.
However, its influence is believed negligible at the strain level of peak and critical stress ratio.• Noting that in monotonic shearing, the direction of force anisotropy is coaxial with the loading direction,
the degree of non-coaxiality can be estimated by Eq. (29). It depends on the ratio between the degrees of
fabric and force anisotropies �/B f1 and the deviation between their directions
(ψ − β
f1
)/2. It is found
that the degree of deformation non-coaxiality becomes significant only if: a) the ratio between the degreeof fabric anisotropy and the degree of force anisotropy is sufficiently large, and b) the principal fabricdirection is sufficiently different from the applied loading direction.
Acknowledgments The work reported in this paper is financially supported by The University of Nottingham, UK. The authorswould like to thank the reviewers for their careful reading and constructive comments that have contributed to the quality of themanuscript.
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use,distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
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